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<p>In this paper, we give for the first time a systematic study of the variance of the distance to the boundary for arbitrary bounded convex domains in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathbb {R}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathbb {R}^3\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>. In dimension two, we show that this function is strictly convex, which leads to a new notion of the centre of such a domain, called the variocentre. In dimension three, we investigate the relationship between the variance and the distance to the boundary, which mathematically justifies claims made for a recently developed algorithm for classifying interior and exterior points with applications in biology.</p>

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Variance of the Distance to the Boundary of Convex Domains in \(\mathbb {R}^{2}\) and \(\mathbb {R}^{3}\)

  • 期刊论文

  • 发表日期：

    2026年2月24日

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  • Alastair N. Fletcher,
  • Alexander G. Fletcher

摘要

In this paper, we give for the first time a systematic study of the variance of the distance to the boundary for arbitrary bounded convex domains in \(\mathbb {R}^2\) R 2 and \(\mathbb {R}^3\) R 3 . In dimension two, we show that this function is strictly convex, which leads to a new notion of the centre of such a domain, called the variocentre. In dimension three, we investigate the relationship between the variance and the distance to the boundary, which mathematically justifies claims made for a recently developed algorithm for classifying interior and exterior points with applications in biology.